Building an Icosahedron with Golden Rectangles
In this activity you will learn to build an icosahedron. Here is one below. Drag the red point to see its net
The Icosahedron is a solid with 20 equilateral triangles as faces. It has 12 vertices and 30 edges. The vertices can be plotted using 'Golden Rectangles'. See the steps build a 'Golden Rectangle' in the window below
Steps for Drawing a Golden Rectangle
1. Draw segment AB from (0,0) to (1,0).
2. Use the Regular Polygon tool to make a square on
segment AB.
3. Use the Midpoint Tool to make the midpoint of
segment AB.
4. Draw ray AB and ray DC.
5. Draw a circle centered at E with radius EC.
6. Mark point F where the circle intersects ray AB.
7. Use the Perpendicular Line Tool to draw
perpendicular line through F to ray DC.
8. Mark the intersection of the line from step 7 with
ray CD. This is point G.
9. Hide the Circle, Square, and Lines. Then use the
polygon tool draw Rectangle ADGF. This is a Golden Rectangle.
Use the distance tool to measure side DG. This measurement is the Golden Ratio. What is it approximately?
A golden rectangle is special because it is self-similar. The small rectangle is proportional to the Biggest one. We can see that it can be rotated and dilated to land on the big one.
Click Here to find the Golden Ratio with a Quadratic Equation
Below is a Golden Rectangle. Perform the rotations described below to construct the icosahedron
Use the Rotate Around Line tool to rotate the rectangle around each dotted line by 90 degrees. Then Hide the original. The resulting location of the corners of each rotated rectangle are the vertices of the icosahedron!